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Let P(z) be a polynomial of degree n having all its zeros in |z|≤ k. For k=1, it is known that for each r> 0 and |α|≥ 1, n(|α|-1){_0^2π|P(e^iθ)|^rdθ}^1r ≤ {_0^2π|1+e^iθ|^rdθ}^1r _|z|=1|D_αP(z)|. In this paper, we shall first consider the case when k≥ 1 and present certain generalizations of this inequality. Also for k≤ 1, we shall prove an interesting result for Lacunary type of polynomials from which many results can be easily deduced. , number=1, journal=Analysis in Theory and Applications, year=2017, month=Jan., pages=81–91